$h=1_{[-1,1]}$ and $g_n=1_{[-n,n]}$. Determine $f_n$ such that $\hat f_n=g_n \star h$

Question

Let $h=1_{[-1,1]}$ and $g_n=1_{[-n,n]},\ n \in \mathbb N$. ($1_{[.,.]}$ denotes the indicator function.)

How can I determine $f_n \in L^1(\mathbb R)$ such that $\hat f_n$ is the convolution of $g_n$ and $h$:

$$\hat f_n=g_n \star h$$

$\hat f_n(x):=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f_n(t)e^{-itx}dt$

$g_n \star h=\int_{-1}^{1}1_{[-n,n]}(x-y) dy=\int_{-1}^{1}1_{[x-n,x+n]}(y) dy=\min(1,x+n)-\max(-1,x-n).$

Answer 1

Since $(g_n\ast h) \in L^1, \forall n \in \Bbb{N},$ then by Fourier inversion consider the functions $$f_n(x)=\int_{\Bbb{R}}(g_n\ast h)(\xi)e^{2 \pi i \xi x}d\xi$$